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Absolute Continuity

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Absolute Continuity on Compact Intervals

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If a function is absolutely continuous on [a, b], then it is continuous on [a, b] and has a derivative ff' almost everywhere on [a, b], such that the integral of ff' over [a, b] equals f(b)f(a)f(b) - f(a).

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Lebesgue's Characterization of Absolute Continuity

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Lebesgue's theorem states that a continuous function of bounded variation on [a, b] is absolutely continuous if and only if its set of discontinuities for its derivative is of Lebesgue measure zero.

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Radon-Nikodym Theorem

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The Radon-Nikodym theorem provides that if μν\mu \ll \nu, then there exists an integrable function ff such that μ(E)=Efdν\mu(E) = \int_E f d\nu for all measurable sets E.

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Absolute Continuity and Variation

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A function is absolutely continuous on [a, b] if and only if it is both continuous and of bounded variation on that interval.

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Relation to Almost Everywhere Differentiability

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An absolutely continuous function on a compact interval is differentiable almost everywhere on that interval.

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Definition of Absolute Continuity of a Function

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A function f is absolutely continuous on an interval [a, b] if for every ϵ>0\epsilon > 0 there exists δ>0\delta > 0 such that for any finite collection of disjoint sub-intervals [xk,yk]{[x_k, y_k]}, the sum of the lengths of the sub-intervals is less than δ\delta implies the sum of f(yk)f(xk)|f(y_k) - f(x_k)| is less than ϵ\epsilon.

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Absolute Continuity vs Uniform Continuity

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A function that is absolutely continuous on an interval is also uniformly continuous on that interval, but the converse is not necessarily true.

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Characterization by Derivatives

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If ff is differentiable almost everywhere on [a, b], has a Lebesgue integrable derivative ff', and f(x)=f(a)+axf(t)dtf(x) = f(a) + \int_{a}^{x} f'(t) dt for all xx in [a, b], then ff is absolutely continuous on [a, b].

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Absolute Continuity of Measures

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A measure μ\mu is absolutely continuous with respect to another measure ν\nu (denoted μν\mu \ll \nu) if ν(E)=0\nu(E) = 0 implies μ(E)=0\mu(E) = 0 for every measurable set E.

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Total Variation of an Absolutely Continuous Function

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The total variation of an absolutely continuous function ff on an interval [a, b] can be computed as the integral of the absolute value of its derivative: Vab(f)=abf(x)dxV_{a}^{b}(f) = \int_{a}^{b} |f'(x)| dx.

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